0,2

Let P = Pascal's triangle as an infinite lower triangular matrix and A = the infinite array of arithmetic sequences as shown in A077028:

1 1 1 1 1...

1 2 3 4 5...

1 3 5 7 9...

1 4 7 10 13...

1 5 9 13 17...

Perform the operation P * A, getting a new array with each column being the binomial transform of an arithmetic sequence. Take antidiagonals of the new array, then by rows = the triangle of A105851.

n-th column of the triangle is the binomial transform of the arithmetic sequence (n*k + 1), (k=0, 1, 2...).

Column 3: 1, 5, 16, 44, 112...(A053220) is the binomial transform of 3k+1 (A016777: 1, 4, 7,...).

Triangle begins:

1;

2, 1;

4, 3, 1;

8, 8, 4, 1;

16, 20, 12, 5, 1;

32, 48, 32, 16, 6, 1;

64, 112, 80, 44, 20, 7, 1;

128, 256, 192, 112, 56, 24, 8, 1;

256, 576, 448, 272, 144, 68, 28, 9, 1;

512, 1280, 1024, 640, 352, 176, 80, 32, 10, 1;

1024, 2816, 2304, 1472, 832, 432, 208, 92, 36, 11, 1 ;...

Cf. A077028, A001792, A001787, A053220, A016777, A014480.

Sequence in context: A055248 A103316 A140069 this_sequence A164967 A106195 A051129

Adjacent sequences: A105848 A105849 A105850 this_sequence A105852 A105853 A105854

nonn,tabl

Gary W. Adamson (qntmpkt(AT)yahoo.com), Apr 23 2005

More terms from Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Mar 31 2007